Scott Aaronson's excellent "Who Can Name the Bigger Number?" should definitely be read by everyone.
Those hungry for followups should see this HN discussion (note the comments by Eliezer) and this one on MathOverflow. Also there's a related problem about specifying big computable numbers, discussed on HN here. Both problems seem to be related to defining large countable ordinals, though a detailed discussion of the relationship is too large for this margin to contain.
upvoted because people's time is much better spent learning about BB than about creative recursion.
(2) is a number that cannot be meaningfully expressed, understood, or calculated by any means that exist today.
You just did meaningfully express it, by giving it a definition. From its definition, we can infer some of its properties. For example, it's even!
Also, we have a standard huge number example used on LW: see Knuth's up-arrow notation.
I was basically aware of that notation; I did not realize how much more elegant it is, in terms of being easy to explain.
Those are really big numbers, but while they start out large, the functions /x\, [x], ..., still grow more slowly than A(x,x), and are pitifully small compared to BB(x). I'm not actually sure where S(x) = {x-sided shape of x} fits in, except that it's computable, so still smaller than BB(x) (for all sufficiently large values of x).
EDIT: I misunderstood the definition of (x) as a circle, and thus as the end of a limiting process. The op intended it to be a polygon with [X] sides. The next paragraph is not valid, although the final one is.
However, (x) is not a number, it's a limit, and more importantly it is infinite for x > 1. Since you were in an analysis class, this should have been talked about!
What I'm really confused by is
Imagine drawing a line from that point to one on the near surface of the sun, which represents infinity. (2) lies within the parentheses surrounding that period, and that's an understatement of how close it is to zero.
Not only is (2) an infinite quantity, if it wasn't it could be wherever you wanted to put it on the line. If you're mapping 0 to infinity onto a finite length line, you can approach the limit any way you desire. Why not put 1 at the halfway mark and 2 at the three-quarter mark? That seems to me to be conceptually most simple. It's not like we can have a 1-1 mapping of numerical increase to distance!
(2) is clearly a finite quantity. I'm not seeing how you can think otherwise unless I've seriously miscommunicated something.
It is; I misunderstood, although I don't think your notation is blameless.
Basically, in the sequence triangle->square->pentagon->... appears to be a process that approaches circle as the limit of the number of sides tends towards infinity. My first (and second) time reading through the article I missed that (x) is not circle of x, but rather the [x]-gon of x.
I don't see what you're trying to say. Finite numbers are different than infinite ones. What do you mean by a second point representing infinity? Is it supposed to be the number of points between it and the period? In that case, ((2)) isn't even on there, or any other finite number besides zero. The paradoxes of infinity aren't because it's big. They're because it doesn't behave the same as finite numbers.
Finite numbers are different than infinite ones.
What's "infinite number" you refer to, for finite numbers to be different from it? (There're infinite cardinals, of course, but that's set theory, a step bigger than flipping the "finite" modifier.)
All of them. Different ones behave in different ways, but none of them behave like finite numbers.
... but that's set theory, a step bigger than flipping the "finite" modifier.
What if I started with finite cardinals?
I think what we'd use for utility is like cardinal numbers, although that's not precisely what it is. There isn't a set of different QALYs. The finite values are more like real numbers. You still talk about how many QALYs, though.
I second Oscar's note below. Incidentally, the operation you are talking about is very similar to (although not precisely the same as) Knuth's arrow notation.
The reference to polygons suggests a variant of Steinhaus-Moser notation, though I think those are pretty similar anyway.
If you stereographically project the real numbers onto the unit circle and use the metric inherited from R^2, then in fact (2) is very close to infinity.
If you use the arctangent to project the positive reals onto a finite set, then again I would guess that (2) is very close to infinity.
There's an old joke...
Some psychologists do an experiment. They put a mathematician and a plumber on one side of a room, and a beautiful woman on the other side. They say, "You can cross half the remaining distance in the room as many times as you like." The mathematician sighs, "I'd never reach her!" The plumber shrugs and says, "You'd get close enough."
I upvoted this post for blowing my mind, and for causing Oscar to post that link, which did it again.
I know almost nothing about infinite sets but if we define X(t) as the sum of the first t twin primes then is it possible that X(t) might be close to infinity or at least a lot bigger than your ((2)) for t equal to, say, one billion?
Edit Edit: It seems I double misinterpreted the parent. I think we've agreed that (2) is finite, so if I ramble about it being infinite below, ignore me.
Being "close to infinity" doesn't really make sense in standard real analysis, if you're talking about any given finite number, because you can always produce another number arbitrarily larger than whatever you wanted to show was "really close to infinity".
Edit: So if you're asking "what's bigger, the sum of the first billion twin primes, or (2)", this question doesn't make sense because (2) isn't finite, but that sum is.
What's even more interesting, though, is that you can meaningfully compare different sized infinities. Look into Cardinality.
EDIT: I misunderstood the op, as can be seen from this post and the child.
I don't understand why no one else is objecting to treating (2) as a number.
If F(x, s) = {s-sided function of x}, e.g. F(2, 3) = /2\, F(2,5) = [2>, then clearly F(2,x) > 2^x for x > 3.
(2) is the limit of F(2, x) as x approaches infinity; just as 2^x is infinite in the limit, so is (2). I'm not even sure whether ((2)) is well-defined, because we haven't been told how it approaches the limits, and it's not clear to me that all methods yield the same function.
Oh actually you're right. I didn't interpret the op correctly. I thought it was just some weird extension of Knuth's up arrow notation but now I see what's going on.
In that sense, (2) isn't a real number, as infinity isn't a real number, it's an extended real number.
And I think you're right again, ((2)) isn't well defined I don't think.
Count me wrong. You understood correctly the first time. See Vladimir's comment; the notation is confusing, but it is a finite process.
(2) is the limit of F(2, x) as x approaches infinity
It's not. As I understand from the post, in your notation, (2)=F(2,[2])=F(2,F(2,4)).
Hmm, now I think you might be right, and that I misunderstood the poster's original intention. The paragraph currently reads
... I'll spare the next [X] operators, and go right to (X) ("circle-X"). (X) follows the process that took us from triangle to square to pentagon, iterated an additional [X] times.
Is that an edit? I do not remember the phrase following the last comma. The notation is at least confusing, in that triangle->square->pentagon->...->circle ought to represent a limiting process, rather than a finite one.
Thank you. I would ask the op to use a less confusing notation, but I will go ahead and edit my other objections.
Being "close to infinity" doesn't really make sense in standard real analysis
Sure it does -- the extended real line is easily made into a metric space.
(The point being that your use of the term "standard real analysis" here is a bit off, specifically in the form of being insufficiently meta. "Real analysis" is a subject in which one considers such ideas as "metrics" (and many other things) in general; the term doesn't just refer narrowly to the properties of the real line equipped with all the most "standard" structures.)
Fair enough, I was simply trying to appeal to what is probably his most familiar intuition regarding the real number line.
Most people that are going to have confusion about a big number being close to infinity probably aren't going to know what a metric space is.
They may not know the term "metric space" (in which case you just explain that it's a setting where we can measure distance), but if they think larger numbers are "closer to infinity" than smaller numbers, that means they are intuitively thinking in terms of the extended real line (metrized in one of the usual ways).
As a ward against any confusion - because I expect at least one person will make this mistake after reading your comment - it should be noted that while as a topological space it's metrizable, the resulting metric on the reals would necessarily have pretty different properties (it would be the same topologically, but not uniformly, e.g.).
[This is from a very neat example my real analysis professor used some years ago. While I'm fairly confident it's neat, I'm not certain it's top-level-post-worthy. The general point is about problems with applying concepts involving infinity to reality; any advice on content (or formatting!) would be greatly welcomed. My math education basically ended after a few upper division courses, so it's possible there are some notational schemes or methods I am ignorant of.
I think this is a fun little exercise, if nothing more.]
The concept of "infinity" and "infinite series" and sets get thrown around a lot in mathematics and some of philosophy. It's worth trying to put the concept of infinity in perspective before we try to think of things in the real world being "infinite." Warning: this post will involve numbers that are literally too large to comprehend. But that's the point.
Let us define an operator, /X\ ("triangle-X"). /X\ = X raised to the X power X times. Thus, /2\ = 22^2 = 24 = 16.
//2\\ ("2-triangle-2") would do this operation twice. Thus, it would equal /16\, the value of which we'll get to in a minute.
We now introduce a new operator, [X] ("square-X"). [X] = triangle-X-triangle-X, i.e. X inside of /X\ triangles. [2] = ////////////////2\\\\\\\\\\\\\\\\
We can introduce another operator, [X> ("pentagon-X"). [X> = X inside of [X] squares. I believe this would be "square-X-square-X").
...
[Edited for clarity]
I'll spare the next [X] operators, and go right to (X) ("circle-X"). Technically, it's whole-lot-of-sides-polygon-X - we could continue this process indefinitely - but we'll call it circle-X, because that's as far as we're going. (X) follows the process that took us from triangle to square to pentagon, iterated an additional [X] times.
I'll be honest. This got kind of meaningless a bit before [X]. Let's start trying to construct what [2] equals, and you'll see why.
/2\=16. So //2\\ = /16\ = 1616^16^16^16^16^16^16^16^16^16^16^16^16^16^16^16. Using some very rough approximations, we can say this is about 102x10^19, or one followed by twenty billion billion zeroes. //16\\ is thus one followed by twenty billion billion zeroes, raised to the power of one followed by twenty billion billion zeroes one followed by twenty billion billion zeroes times. My math education could be more complete, but I am not aware of another way to denote such a number. To say it could not be written in scientific notation on a universe-sized sheet of paper is probably a colossal understatement. And after we calculate that number we have to repeat the process thirteen more times to get [2]. We could theoretically keep doing this until we got to (2); (2) is a number that cannot be meaningfully expressed, understood, or calculated by any means that exist today. And there's still ((2)) after that.
Now, imagine that this period (.) represents zero. Imagine drawing a line from that point to one on the near surface of the sun, which represents infinity (yes, this is improper - it's a finite line - but the point is visualization, so understatement really isn't an issue here). (2) lies within the parentheses surrounding that period, and that's an understatement of how close it is to zero. It really isn't even 1/(2) inches from that period.
Remember this the next time you ponder the meaning, use, or existence of an infinite set, infinite repetitions, or an infinite time.